Characterization of 2-Local Derivations and Local Lie Derivations on Some Algebras

We prove that each 2-local derivation from the algebra M_n(A ) (n > 2) into its bimodule M_n(M) is a derivation, where A is a unital Banach algebra and M is a unital A -bimodule such that each Jordan derivation from A into M is an inner derivation, and that each 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, the Jiang–Su algebra, and UHF algebras.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

A solution to the Bézout equation in <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) without Gelfand theory

Let \({M\subseteq \mathbb{C}}\) be compact and \({K\subseteq M}\) closed, and let A(K, M) be the uniform algebra of all functions continuous on M and holomorphic in the interior K° of K. We present a constructive proof of Arens’ classical result that for \({(f_{1},\ldots,f_{n})\in A(K,M)^{n}}\) the Bézout equation \({\sum_{j=1}^{n} a_{j}f_{j}=1}\) has a solution in A(K, M) if and only if the functions f _j have no common zero in K. We shall also consider matrix-valued Bézout equations.     

Inner derivations of simple Lie pencils of rank 1

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M ₃(U,D′), where U is the subspace of all skew-symmetric matrices in M ₃(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M ₃(P).     

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras \(\mathbb{A}\) _m with unity. The following statements are proved.If a holomorphic linear connection ? on M _n over \(\mathbb{A}\) _m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ? of the Lie algebra of all affine vector fields of the space (M _mn ^? , ?^?) is no greater than (mn)² ? 2mn + 5, where m = dim_? \(\mathbb{A}\), \(n = dim_\mathbb{A} \) M _n , and ?^? is the real realization of the connection ?.Let ?^? =¹ ? ײ ? be the real realization of a holomorphic linear connection ? over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor ¹ R ≠ 0, ² R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M _2n ^? , ?^?) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( ^a M _n , ^a ?) (a = 1, 2).     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

Strictly nontame primitive elements of the free metabelian Lie algebra of rank 3

We prove that the free metabelian Lie algebra M ₃ of rank 3 over an arbitrary field K admits strictly nontame primitive elements.     

A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature n- 1. By developing some new techniques, Colding(1996) proved that the following three conditions are equivalent: 1)dGH(M, S~n) → 0; 2) the volume of M Vol(M) → Vol(S~n); 3) the radius of M rad(M) →π. By developing a different technique, Petersen(1999) gave the 4th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λ_(n+1)(M) → n, is also equivalent to the radius of M, rad(M) →π, and hence the other two.In this paper, we use Colding's techniques to give a new proof of Petersen's theorem. We expect our estimates will have further applications.     

The extrinsic holonomy Lie algebra of a parallel submanifold

We investigate parallel submanifolds of a Riemannian symmetric space N. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full, complete, parallel submanifolds of N, usually called “1-fullness” of M. Furthermore, for every parallel submanifold \({M\subset N}\) we consider the pullback bundle T N| _M with the linear connection induced by \({\nabla^N}\) . Then there exists a distinguished parallel subbundle \({\mathcal {O}M}\) , usually called the “second osculating bundle” of M. Given a parallel isometric immersion from a symmetric space M into N, we can describe the “extrinsic” holonomy Lie algebra of \({\mathcal {O} M}\) by means of the second fundamental form and the curvature tensor of N at some fixed point. If moreover N is simply connected and M is even a full symmetric submanifold of N, then we will calculate the “extrinsic” holonomy Lie algebra of T N| _M in an explicit form.     

Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case

We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations f _t + cf _x + F(t, x, c)f _c = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.     

Degenerations of Submodules and Composition Series

Let M and N be modules over an artin algebra such that M degenerates to N. We show that any submodule of M degenerates to a submodule of N. This suggests that a composition series of M will in some sense degenerate to a composition series of N. We then study a subvariety of the module variety, consisting of those representations where all matrices are upper triangular. We show that these representations can be seen as representations of composition series, and that the orbit closures describe the above mentioned degeneration of composition series.     

Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E₂. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.     

Superization of homogeneous spin manifolds and geometry of homogeneous supermanifolds

Let M ₀=G ₀/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition \(\mathfrak {g}_{0}=\mathfrak {h}+\mathfrak {m}\) and let S(M ₀) be the spin bundle defined by the spin representation \(\tilde{ \operatorname {Ad}}:H\rightarrow \mathrm {GL}_{\mathbb {R}}(S)\) of the stabilizer H. This article studies the superizations of M ₀, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to the sheaf of sections of Λ(S ^*(M ₀)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry \(\mathfrak {g}_{0}\) to an algebra of supersymmetry \(\mathfrak {g}=\mathfrak {g}_{\overline {0}}+\mathfrak {g}_{\overline {1}}=\mathfrak {g}_{0}+S\) via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection \(\nabla^{\mathcal {S}}\) in the spin bundle S(M ₀). Killing vectors together with generalized Killing spinors (i.e. \(\nabla^{\mathcal {S}}\)-parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection \(\nabla^{\mathcal {S}}\) defines a superconnection on M, via the super-version of a theorem of Wang.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.     

Group analysis of the one-dimensional Boltzmann equation: I. symmetry groups

We consider the one-dimensional Boltzmann equation f _t + cf _x + Ff _c = 0, where the functions f and F are assumed to depend on three variables t, x, and c. We obtain relations defining the symmetry algebra in the general case and also under the additional conditions of conservation of the relations dx = c dt and dc = F dt, which arise from physical considerations. We show that the widest symmetry algebra is obtained in the case of conservation of both relations. This algebra is infinite-dimensional, and its structure is independent of the form of the function F.